The classic "Limit Dislribntions fOT slt1ns of Independent Ramdorn Vari ables" by B.V. Gnedenko and A.N. Kolmogorov was published in 1949. Since then the theory of summation of independent variables has devel oped rapidly. Today a summing-up of the studies in this area, and their results, would require many volumes. The monograph by I.A. Ibragi mov and Yu. V. I~innik, "Independent and Stationarily Connected VaTiables", which appeared in 1965, contains an exposition of the contem porary state of the theory of the summation of independent identically distributed random variables. The present book borders on that of Ibragimov and Linnik, sharing only a few common areas. Its main focus is on sums of independent but not necessarily identically distri buted random variables. It nevertheless includes a number of the most recent results relating to sums of independent and identically distributed variables. Together with limit theorems, it presents many probahilistic inequalities for sums of an arbitrary number of independent variables. The last two chapters deal with the laws of large numbers and the law of the iterated logarithm. These questions were not treated in Ibragimov and Linnik; Gnedenko and KolmogoTOv deals only with theorems on the weak law of large numbers. Thus this book may be taken as complementary to the book by Ibragimov and Linnik. I do not, however, assume that the reader is familiar with the latter, nor with the monograph by Gnedenko and Kolmogorov, which has long since become a bibliographical rarity

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The series is devoted to the publication of high-level monographs and surveys which cover the whole spectrum of probability and statistics. The books of the series are addressed to both experts and advanced students.

It has been a rare privilege to assemble this volume of Wassily Hoeffding's Collected Works. Wassily was, variously, a teacher, supervisor and colleague to us, and his work has had a profound influence on our own. Yet this would not be sufficient reason to publish his collected works. The additional and overwhelmingly compelling justification comes from the fun damental nature of his contributions to Statistics and Probability. Not only were his ideas original, and far-reaching in their implications; Wassily de veloped them so completely and elegantly in his papers that they are still cited as prime references up to half a century later. However, three of his earliest papers are cited rarely, if ever. These include material from his doctoral dissertation. They were written in German, and two of them were published in relatively obscure series. Rather than reprint the original articles, we have chosen to have them translated into English. These trans lations appear in this book, making Wassily's earliest research available to a wide audience for the first time. All other articles (including those of his contributions to Mathematical Reviews which go beyond a simple reporting of contents of articles) have been reproduced as they appeared, together with annotations and corrections made by Wassily on some private copies of his papers. Preceding these articles are three review papers which dis cuss the . impact of his work in some of the areas where he made major contributions.

High-dimensional probability offers insight into the behavior of random vectors, random matrices, random subspaces, and objects used to quantify uncertainty in high dimensions. Drawing on ideas from probability, analysis, and geometry, it lends itself to applications in mathematics, statistics, theoretical computer science, signal processing, optimization, and more. It is the first to integrate theory, key tools, and modern applications of high-dimensional probability. Concentration inequalities form the core, and it covers both classical results such as Hoeffding's and Chernoff's inequalities and modern developments such as the matrix Bernstein's inequality. It then introduces the powerful methods based on stochastic processes, including such tools as Slepian's, Sudakov's, and Dudley's inequalities, as well as generic chaining and bounds based on VC dimension. A broad range of illustrations is embedded throughout, including classical and modern results for covariance estimation, clustering, networks, semidefinite programming, coding, dimension reduction, matrix completion, machine learning, compressed sensing, and sparse regression.

This classroom-tested textbook is an introduction to probability theory, with the right balance between mathematical precision, probabilistic intuition, and concrete applications. Introduction to Probability covers the material precisely, while avoiding excessive technical details. After introducing the basic vocabulary of randomness, including events, probabilities, and random variables, the text offers the reader a first glimpse of the major theorems of the subject: the law of large numbers and the central limit theorem. The important probability distributions are introduced organically as they arise from applications. The discrete and continuous sides of probability are treated together to emphasize their similarities. Intended for students with a calculus background, the text teaches not only the nuts and bolts of probability theory and how to solve specific problems, but also why the methods of solution work.